A math circle...

I mentioned Pythagorean triples to my pre-calc students last month, and told them if enough of them were interested, I would run a math circle on this topic. Ten of them signed up, six came the first week, and three came to the second and third sessions. It's small, but it's going very well, and I may do something bigger next semester.

This is my first time doing an ongoing math circle with many sessions devoted to one topic. It's also my first time getting my own students to come to a math circle. I am really happy that they keep coming. I originally said it would be five sessions, but I can see that we could easily go for six to eight sessions on the questions raised here. (I may let them talk me into extending it.)

I love it when the way they approach a problem is different than the way I would have done it. X saw a pattern I had not seen before, and we explored her pattern at length in the second session. I haven't had time to write up the details, and have probably forgotten much of it.


Week One
What examples can we come up with?  (3-4-5, 5-12-13, ...)
6-8-10 leads us to define primitive Pythagorean triples (in which gcf(a,b,c)=1; 6-8-10 isn't primitive)
Maybe writing a list of all the perfect squares up to 400 will help us find more.
What patterns do we see?

• Odd + Even = Odd
• Middle number is a multiple of 4
• c = b+1 (after which I added 8-15-17 to our list)

Week Two
One person was new, so we reviewed our first week's work for him.

We explored the "family" of triples with c = b+1. a² + b² = (b+1)² becomes a² = (b+1)² - b²
= b² + 2b + 1 - b² = 2b-1. If  a² = 2b-1, then b = (a²+1)/2. This will be a whole number whenever a is an odd number. So we got lots more: 7-24-25, 9-40-41, ...

X noticed that in the triples

3-4-5
5-12-13
7-24-25
9-40-41

the second number is 4*1, 4*3, 4*6, 4*10, ... For the nth one, we use 4 times a number n more than the previous one. I showed them why these (1, 3, 6, 10, ...) are called triangle numbers, and asked them to add 1 to 100. They each came up with their own way of thinking about it. We came back to X's pattern and wrote:

a=2n+1
b=4*n(n+1)/2=2n(n+1)
c=b+1

Week Three
Another new person came, so we summarized for her. Then we explored triples where c = b+2.

I love seeing their creativity and persistence. At the same time, I am blown away by the holes in their understanding of algebra moves. Y was considering (4n)², and thought he might have to distribute.

We verified that we get all of the primitive Pythagorean triples with c=b+2 using:

a=4n
b=4n²-1
c=4²+1

Not sure where we'll take it in Week Four, but eager to find out. I am still struggling to lead less, become less visible, and listen more.